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# Prove the Wronskian satisfies a first-order differential equation

The Wronskian is defined by

for given functions and .

Let be the Wronskian of two solutions and of the differential equations

where and are constants.

1. Prove that satisfies the first-order linear differential equation

and hence,

By this formula we can see that if then for all .

2. Assume is not identically zero and prove that if and only if is constant.

1. First, we evaluate where is the Wronskian of the two functions and .

Furthermore, by Theorem 8.3 (page 310 of Apostol), since is a solution of we know

since . Hence, if .

2. Assume . Then for all . By part (a) of the previous exercise (Section 8.14, Exercise #21) we know is constant.

Conversely, assume is constant. Then, again by the previous exericse, we have for all . Hence,

### One comment

1. Anonymous says:

In part b), why it is said that if W(0) = 0, then W(x) = 0 for all x?